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Hello there!
Well first: In Pathfinder HD means something like level. A druid of level 8 has 8HD because he hitpoints of 8 dice-rolls (well, 7 assuming the first one is always maximum)
So when an animal companion starts with 2 HD it kind of is level 2. when you look at the BAB of a druid you will see he's at +1 there as well.
Now when the companion has 3HD it gained a level, so you have to toll one die to determin how many hitpoints it adds to its total. Sometimes companions do not gain a level when the druid gains one. Thats when the HD in that table does not change. The companion does not gain new hitpoints or similar level dependent stuff (like feats, stat boosts or saving throw increases)
In the end it caps at 16 HD. when you compare it to the druid table you will see that the druid has a BAB of 12 at level 16, so when the druid is level 20 the animal companion has 16 HD (=is level 16) and the BAB of a level 16 druid: 12.
Claws and bites are primary natural attacks. That means all of them are done at full BAB and all of them add the full strength modifier to damage.
Rake means that as soon as the companion grapples (what it can always do, check the rules for a grapple maneuver) it can deliver these rake attacks.
In addition to that a big cat has the grab special attack that allows to start a grapple as free action after you hit with the appropriate natural attack.
Further you can also use the rake special attack when using the allmighty pounce special attack. Then you end up having 5 attacks at full attackbonus and strenth bonus to damage after a charge.
Wasum!
1) A 2 lvl druid has +1 bab - the animal starts with 2 HD = +1 bab
2) At lvl 3 the animal goes from 2 to 3 HD - so you add 1d8 to the 7 it already has.
3) you treat them as primary natural attacks. Sp full bab and 1x str.
have a look at :
Natural attacks
4) Ninjaed!
1) A 2 lvl druid has +1 bab - the animal starts with 2 HD = +1 bab
2) At lvl 3 the animal goes from 2 to 3 HD - so you add 1d8 to the 7 it already has.
3) you treat them as primary natural attacks. Sp full bab and 1x str.
have a look at :
Natural attacks4) Ninjaed!
That was more "turtled":P
Greetings, fellow traveller.
Glad to hear another player fell for the druid!
The PRD (Player Reference Document, which you can find as a link at the left right margin of this website (right under the search button) is a really great resource for all things concerning Pathfinder.
There you'll find a table under the Class/Druid section which depicts the career of your (girlfriend's) druid and if you scroll down, you'll get to the Animal Companion (AC) subsection.
There, another table awaits you giving you the same convinient overview about your animal companion's career.
All the entries are totals; e. g. your saving throw bonus for your Fortitude saving throw is a flat +3 bonus at levels 1, 2, and 3 for your AC. If it reaches you add +1 to it, giving you the +4 total as stated in the table.
The same goes for all the other entries.
All these entries are the basic numbers, which might then be modified by the AC's attribute modifiers (e. g. the Fortitude saving throw is not only level dependant, but is also modified by the AC's Constitution bonus - for a cat (big) that'll be +1 due to it having CON 13).
Even further down, you'll find the exclusive statistics for every single AC a druid can pick (which might differ from what the Bestiary entry gives for the animal). If you have a look at the Cat, big entry you'll notice that it evolves at reaching level 7: it gets bigger and more ferocious.
Under special attacks it gets pounce and grab (and rake).
Now all of the attacks of your AC can be made with an additional attempt to grapple its oponent (via the grab special attack. You can now easily set up the condition needed for your rake attacks.
Ruyan.
Correct, the +8 are its attack bonus d20+8 results in the armor class it hits (equal or higher let's you succeed).
Ruyan.
Hang on - the +8 do not have anything to do with the enemy being grappled. Its just the attack bonus of the rakes, strength modifier + base attack bonus + feats, magic and stuff.
Usually thats the same as the bonus for your claw/bite attack unless you've not taken weaponfocus bite or similar effects that change the attack bonus of a certain attack.
Correct as well (loophole: I didn't mention it being easier to hit a grappled enemy...).